Mathematical Problems in EngineeringVolume 2012 (2012), Article ID 703474, 26 pagesdoi:10.1155/2012/703474

Research Article

The Integrability and Existence of Periodic Solutions on a First-Order Nonlinear Differential Equation with a Polynomial Nonlinear Term

Ni Hua and Tian Li-Xin

Faculty of Science, Jiangsu University, Jiangsu, Zhenjiang 212013, China

Received 18 July 2011; Revised 19 October 2011; Accepted 23 October 2011

Academic Editor: Oded Gottlieb

Copyright © 2012 Ni Hua and Tian Li-Xin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with a first-order differential equation with a polynomial nonlinear term. The integrability and existence of periodic solutions of the equation are obtained, and the stability of periodic solutions of the equation is derived.

1. Introduction

Consider the following first-order nonlinear differential equation: when , (1.1) becomes Ricatti’s equation, when , (1.1) becomes the following nonlinear Abel type differential equation: The nonlinear Abel type differential equation plays an important role in many physical and technical applications [1–9]. The mathematical properties of (1.2) have been intensively investigated in the mathematical and physical literature. Matsuno [10] analyzed a two-dimensional dynamical system associated with Abel nonlinear equation. Strobel and Reid [11], Reid and Strobel [12] have obtained superposition rules (prescriptions for combining a finite number of known particular solutions in such a way to obtain the general solution to a (system of) differential equation(s) without operation of integration) for the Abel type equation, involving four or two particular solutions. Mak et al. [13], Mak and Harko [14] have presented a solution-generating technique for Abel type ordinary differential equation, both suppose that is a particular solution of (1.2), by means of the transformations methods, and present an alternative method of generating the general solution of (1.2) from a particular one.

Zheltukhin and Trzetrzelewski [15] developed the geometric approach to study the dynamics of U(1)-invariant membranes. The approach reveals an important role of the Abel nonlinear differential equation of the first type with variable coefficients depending on time and one of the membrane extendedness parameters. The general solution of the Abel equation was constructed.

However, little work was done about the integrability and periodicity of (1.1). In this paper, we discuss the integrability and the periodic solutions of (1.1); the sufficient conditions which guarantee the integrability and the existence of the periodic solutions for (1.1) are obtained, and the stability of the periodic solutions of (1.1) is discussed. To the best of authors’ knowledge, this is the first paper considering the three periodic solutions of (1.1), some new results are obtained.

The present paper is organized as follows. In Section 2, we give three lemmas to be used later. In Section 3, the integrability of (1.1) is derived. In Section 4, the existence and stability of the periodic solutions of (1.1) are obtained. In Section 5, we conclude our results.

2. Preliminary Lemmas

For the sake of convenience, suppose that is a continuous -periodic function defined on , we denote and ; .

Consider the following:, , , are continuous functions defined on .

Lemma 2.1. The domain is positive invariant with respect to (2.1).

Proof. By (2.1), it follows that the assertion is valid for all . The proof is completed.

Lemma 2.2. The domain is negative invariant with respect to (2.1).

Lemma 2.3. Consider the following: , are continuous -periodic functions, if , then (2.3) exists a unique -periodic solution , and can be written as follows: or

Proof. The proof of Lemma 2.3 is a few, such as that of the papers [16, 17] and others, But in [16], the author only proved the case of the almost periodic equation; in [17], the authors just proved that Lotka-volterra equation has a unique globally attractive periodic solution, but they did not give the very two expressions of the periodic solution as above. In order to make the following proof clear, here we give our proof.

From (2.3), it is easy for us to get the unique solution with the initial value which can be written as follows: thus we have Let , by (2.6) and (2.7), if is an -periodic function, then it follows that , thus we can get Substitute (2.8) into (2.6), it follows that

When , it follows that

thus we have

Hence, is a unique -periodic solution of (2.3); we rewrite it as follows:

Similarly, we can prove that when , (2.3) has a unique -periodic solution as follows:

Remark 2.4. Under the conditions of Lemma 2.3, all the infinite integrals above are convergent.

3. A Sufficient Condition of Integrability of the First-Order Nonlinear Differential Equation

In this section, we discuss the integrability of (1.1)

Theorem 3.1. Consider (1.1), where , are continuous functions defined on certain interval , if there is a constant such that the following conditions hold: (), (), , then (i) the general solution of (1.1) can be written as follows: where is an integration constant which makes the number of root greater than zero if is an odd number. (ii) Equation (1.1) has a particular solution

Proof. (i) Since the conditions (), () hold, (1.1) can be written as follows: Let , (3.3) can be changed as follows: This is the Bernoulli-type equation, set , the Bernoulli-type equation (3.4) becomes according to the formula of general solution of linear differential equation, we can get that the general solution of (3.5) is given by By the variable transformation of , it follows that since , we can get Remark 3.2. If is an even number, take “+” in the front of above-corresponding formulas, if is an odd number, take “±”. (ii) By (1.1), we set , thus is continuous on , is also continuous on , hence (1.1) satisfies the principle of existence and uniqueness of differential equation, therefore, (1.1) has no singular solution.From the variable transformation of , it requires , but from (3.4), we know is indeed a solution of (3.4). Since , substitute into (1.1), it just satisfies (1.1), so it is a particular solution of (1.1).This is the end of the proof of Theorem 3.1.

Corollary 3.3. Consider Abel type differential equation (1.2), , are continuous functions defined on certain interval ,  if there is a constant such that the following conditions hold: (), (),then the general solution of (1.2) can be written as follows: and is a particular solution of (1.2).

Corollary 3.4. Consider the following Ricatti’s equation: , are continuous functions defined on certain interval , if there is a constant such that the following condition holds: (),then the general solution of (3.10) can be written as follows: and is a particular solution of (3.10).

4. The Existence and Stability of Periodic Solutions of the First-Order Nonlinear Differential Equation

Define

where , , are the same function and numbers as the following Theorem 4.1, and is any given initial time of (1.1).

Theorem 4.1. Consider (1.1), is an odd number, and , , are continuous -periodic functions defined on , if there is a constant such that the following conditions hold: (), (), ,(), ,(), then (i) equation (1.1) has three -periodic solutions , , , and they can be written as follows: (ii) if given any initial value , then the periodic solution of (1.1) is globally attractive in ; (iii) if given any initial value , then the periodic solution of (1.1) is globally attractive in .

Proof. (i) Since the conditions (), () hold, (1.1) can be written as follows: Let , (4.3) can be changed as follows: This is the Bernoulli-type equation, set , the Bernoulli-type equation (4.4) becomes By (), (), it follows that thus is an -periodic function, therefore, is also an -periodic function and since , (not identically equal to zero), thus we have according to Lemma 2.3, it follows that (4.5) has a unique -periodic solution as follows from (4.9), it is easy to know that , since is periodic on , it is bounded on , so it is positive and bounded, since , and by Lemmas 2.1 and 2.2, it follows that (4.4) has two -periodic solutions as follows: thus is also positive and bounded, and is negative and bounded, by virtue of , (1.1) has two -periodic solutions as follows: and , , , .By (1.1), we set , thus is continuous on , and is also continuous on , hence (1.1) satisfies the principle of existence and uniqueness of differential equation, therefore, (1.1) has no singular solution.From the variable transformation of , it requires , but from (4.4), we know is indeed a solution of (4.4), since , hence is a periodic solution of (1.1). Substitute into (1.1), it just satisfies (1.1), so is indeed a periodic particular solution of (1.1), therefore, we have proved that (1.1) has three -periodic solutions , , and . (ii) According to the theories of linear differential equation, we know that the unique solution of (4.5) with positive initial value is given by Since thus is an -periodic function on , and by (4.7), we know , thus . Following we prove that there is a positive number , such that as , here is any small positive number and .From (4.12), given any positive number , by the condition of Theorem 4.1, it follows that ; following we first prove that has upper bound on According to Lemma 2.3, it follows that So has upper bound on .Secondly, we prove that as Since , for all , there must be , when , we have , hence when , we can get therefore, we have proved that there is a positive number , as , it follows that under the transformation of , , we can get the following.If given the initial value then the unique solution of (1.1) is given by if given the initial value then the unique solution of (1.1) is given by and it is easy to know and , respectively.Define a Lyapunov function as follows: where is the unique solution with the positive initial value of (4.5), and is the unique positive -periodic solution of (4.5), differentiating both sides of (4.24) along the solution of (4.5), we get by (4.24), we have thus by (4.24), we can get by (4.28), that it follows since it follows that substitute (4.31) into (4.29), it follows that According to mean value theorem, we can get where or or Since is positive and bounded, and from (4.19), (4.31), we know that are positive and bounded as , hence , are positive and bounded as , so we have by (4.8), it follows that as , and by (4.38), we can get that the -periodic solution of (1.1) is globally attractive in .(iii) The proof of the periodic solution of (1.1) being globally attractive in is similar to that of the periodic solution of (1.1) being globally attractive in , so we omit it here.This is the end of the proof of Theorem 4.1.

Remark 4.2. In Theorem 4.1, since the transformation of , it follows , thus we suppose the initial value .

Theorem 4.3. Consider (1.1), is an even number, and , , are continuous -periodic functions defined on , if there is a constant such that the following conditions hold: (),(), ,(), ,(), , then (i) equation (1.1) has two -periodic solutions , , and they can be written as follows: (ii) If given any initial value , then the periodic solution of (1.1) is globally attractive in .

Proof. The proof of Theorem 4.3 is similar to that of Theorem 4.1, so we omit it here.

Theorem 4.4. Consider (1.1), is an even number, and , , are continuous -periodic functions defined on , if there is a constant such that the following conditions hold: (),(), ,(), ,(), , then (i) equation (1.1) has two -periodic solutions , , and they can be written as follows: (ii) If given any initial value , then the periodic solution of (1.1) is globally attractive in .

Proof. The proof of Theorem 4.4 is similar to that of Theorem 4.1, so we omit it here.From Theorem 4.1, if , it is easy for us to draw the following corollary.Define where , , are the same function and numbers as the following Corollary 4.5, and is any given initial time of (1.2).

Corollary 4.5. Consider Abel type differential equation (1.2), , are -periodic continuous functions defined on , if there is a constant such that the following conditions hold: (); (); (), ;(); then (i) equation (1.2) has three -periodic solutions , , , and they can be written as follows: (ii) If given any initial value , then the periodic solution of (1.2) is globally attractive in . (iii) If given any initial value , then the periodic solution of (1.2) is globally attractive in .

From Theorems 4.3 and 4.4, if , it is easy for us to draw the following corollaries.

Corollary 4.6. Consider Ricatti’s equation (3.10), , are -periodic continuous functions defined on , if there is a constant such that the following conditions hold: (); (), ;(), ; then (i) equation (3.10) has two -periodic solutions , , and they can be written as follows: (ii) If given any initial value , then the periodic solution of (3.10) is globally attractive in .

Corollary 4.7. Consider Ricatti’s equation (3.10) , are -periodic continuous functions defined on , if there is a constant such that the following conditions hold: (); (), ;(), ; then, (i) equation (3.10) has two -periodic solutions , , and they can be written as follows: (ii) If given any initial value , then the periodic solution of (3.10) is globally attractive in .

Theorem 4.8. Consider (1.1), is an odd number, and , , are continuous -periodic functions defined on , if there is a constant such that the following conditions hold: (); (), ;(), ;(); then, (i) equation (1.1) has three -periodic solutions , , , and they can be written as follows: (ii) if given any initial value , then the periodic solution of (1.1) is globally attractive in , where

Remark 4.9. Where above is the same function as that of the following formula (4.54), and is any given initial time of (1.1).

Proof. (i) Since the conditions (), () hold, (1.1) can be written as follows: Let , (4.48) can be changed as follows: This is the Bernoulli-type equation, set , the Bernoulli-type equation (4.49) becomes By (), (), it follows that thus is an -periodic function, therefore, is also an -periodic function and since , (not identically equal to zero), thus we have according to Lemma 2.3, it follows that (4.50) has a unique -periodic solution as follows: from (4.54), it is easy to know that , since is periodic on , it is bounded on , so it is positive and bounded, since , and by Lemmas 2.1 and 2.2, it follows that (4.49) has two -periodic solutions as follows: thus is also positive and bounded, and is negative and bounded, by virtue of , (1.1) has two -periodic solutions as follows: By (1.1), we set , thus is continuous on , and is also continuous on , hence (1.1) satisfies the principle of existence and uniqueness of differential equation, therefore, (1.1) has no singular solution. From the variable transformation of , it requires , but from (4.49), we know is indeed a solution of (4.49), since , hence is a periodic solution of (1.1). Substitute into (1.1), it just satisfies (1.1), so is indeed a periodic particular solution of (1.1), therefore, we have proved that (1.1) has three -periodic solutions , and . (ii) According to the theories of linear differential equation, we know that the unique solution of (4.50) with the positive initial value is given by If given the initial value , then .Remark 4.10. Since the transformation of , it requires , in addition, as , that is to say, it requires .Under the transformation of , , we can get the following.If given the initial value then the unique solution of (1.1) is given by and it is easy for us to know that .If given the initial value then the unique solution of (1.1) is given by it is easy for us to know that .Therefore, we can draw the following conclusion.If given the initial value , then .Next, we prove that the periodic solution of (1.1) is globally attractive in if given the initial value . by (4.53), it follows that as , from (4.63), we can get that the -periodic solution of (1.1) is globally attractive in .This is the end of the proof of Theorem 4.8.

Theorem 4.11. Consider (1.1), is an even number, and , , are continuous -periodic functions defined on , if there is a constant such that the following conditions hold: (); (), ;(), ;(), ; then (i) equation (1.1) has two -periodic solutions , , and they can be written as follows: (ii) if given any initial value , then the periodic solution of (1.1) is globally attractive in .

Proof. The Proof of Theorem 4.11 is similar to that of Theorem 4.8, so we omit it here.

Theorem 4.12. Consider (1.1), is an even number, and , , are continuous -periodic functions defined on , if there is a constant such that the following conditions hold: (); (), ;(), ;(), ; then (i) equation (1.1) has two -periodic solutions , and they can be written as follows: (ii) if given any initial value , then the periodic solution of (1.1) is globally attractive in .

Proof. The Proof of Theorem 4.12 is similar to that of Theorem 4.8, so we omit it here.

From Theorem 4.8, if , it is easy for us to draw the following corollary.

Corollary 4.13. Consider Abel type differential equation (1.2), , are -periodic continuous functions defined on , if there is a constant such that the following conditions hold: (); (); (), ;(); then (i) equation (1.2) has three -periodic solutions , , , and they can be written as follows: (ii) If given any initial value , then the periodic solution of (1.2) is globally attractive in , where

From Theorem 4.11 and 4.6, if , it is easy for us to draw the following corollaries.

Corollary 4.14. Consider Riccati’s equation (3.10), , are -periodic continuous functions defined on , if there is a constant such that the following conditions hold: (); (), ;(), ; then (i) equation (3.10) has two -periodic solutions , and they can be written as follows: (ii) If given the initial value , then the periodic solution of (3.10) is globally attractive in .

Corollary 4.15. Consider Riccati’s equation (3.10), , are -periodic continuous functions defined on , if there is a constant such that the following conditions hold: (); (), ;(), ; then (i) equation (3.10) has two -periodic solutions , , and they can be written as follows: (ii) If given any initial value , then the periodic solution of (3.10) is globally attractive in .

5. Concluding Remarks

Generally, just as Ricatti’s equation, (1.1) is nonintegrable, however in the present paper, we have got a sufficient condition which guarantees the integrability and have presented some exact periodic solutions of nonlinear equation if the coefficients of (1.1) satisfy certain conditions; moreover, we get that the periodic solutions are globally attractive in certain conditions. These conclusions will have certain applications.

Acknowledgments

The authors are highly grateful to the reviewers for their constructive comments. This work is supported by the National Nature Science Foundation of China (no. 11171135) and Jiangsu Province Innovation Project of Graduate Education (2011, no. 1221190037).

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